2 BRANDEN FITELSON

Confirmation consists in positive probabilistic relevance, and disconfirma- • tion consists in negative probabilistic relevance (where the salient probabil- ities are inductive in the Keynesian [21] sense). GOODMAN’S “NEW RIDDLE” Universal generalizations are confirmed by their positive instances and dis- • BRANDEN FITELSON confirmed by their negative instances. Hempel [17] offers a precise, logical reconstruction of Nicod’s naïve instantial ac- Abstract. First, a brief historical trace of the developments in confirmation the- count. There are several peculiar features of Hempel’s reconstruction of Nicod. I ory leading up to Goodman’s infamous “grue” paradox is presented. Then, Good- will focus presently on two such features. First, Hempel’s reconstruction is com- man’s argument is analyzed from both Hempelian and Bayesian perspectives. pletely non-probabilistic (we’ll return to that later). Second, Hempel’s reconstruc- A guiding analogy is drawn between certain arguments against classical deduc- tive logic, and Goodman’s “grue” argument against classical inductive logic. The tion takes the relata of Nicod’s confirmation relation to be objects and universal upshot of this analogy is that the “New Riddle” is not as vexing as many com- statements, as opposed to singular statements and universal statements. In mod- mentators have claimed (especially, from a Bayesian inductive-logical point of ern (first-order) parlance, Hempel’s reconstruction of Nicod can be expressed as: view). Specifically, the analogy reveals an intimate connection between Good- man’s problem, and the “problem of old evidence”. Several other novel aspects (NC0) For all objects x (with names x), and for all predicate expressions φ and ψ: of Goodman’s argument are also discussed (mainly, from a Bayesian perspective). x confirms [( y)(φy ψy)\ iff [φx & ψx\ is true, and ∀ ⊃ 1 x disconfirms [( y)(φy ψy)\ iff [φx & ψx\ is true. ∀ ⊃ ∼ 1. Prehistory: Nicod & Hempel As Hempel explains, (NC0) has absurd consequences. For one thing, (NC0) leads to a theory of confirmation that violates the hypothetical equivalence condition: In order to fully understand Goodman’s “New Riddle,” it is useful to place it (EQC ) If confirms , then confirms anything logically equivalent to . in its proper historical context. As we will see shortly, one of Goodman’s central H x H x H aims (with his “New Riddle”) was to uncover an interesting problem that plagues To see this, note that — according to (NC0) — both of the following obtain: Hempel’s [17] theory of confirmation. And, Hempel’s theory was inspired by some a confirms “( y)(Fy Gy),” provided a is such that Fa & Ga. of Nicod’s [25] earlier, inchoate remarks about instantial confirmation, such as: • ∀ ⊃ Nothing can confirm “( y)[(Fy & Gy) (Fy & Fy)],” since no object Consider the formula or the law: A entails B. How can a particular propo- • ∀ ∼ ⊃ ∼ a can be such that Fa & Fa. sition, [i.e.] a fact, affect its probability? If this fact consists of the pres- ∼ ence of B in a case of A, it is favourable to the law . . . on the contrary, if it But, “( y)(Fy Gy)” and “( y)[(Fy & Gy) (Fy & Fy)]” are logically consists of the absence of B in a case of A, it is unfavourable to this law. ∀ ⊃ ∀ ∼ ⊃ ∼ equivalent. Thus, (NC0) implies that a confirms the hypothesis that all Fs are Gs By “is (un)favorable to,” Nicod meant “raises (lowers) the inductive probability of”. only if this hypothesis is expressed in a particular way. I agree with Hempel that this

In other words, Nicod is describing a probabilistic relevance conception of confir- violation of (EQCH ) is a compelling reason to reject (NC0) as an account of instantial mation, according to which it is postulated (roughly) that positive instances are confirmation.2 But, I think Hempel’s reconstruction of Nicod is uncharitable on probability-raisers of universal generalizations. Here, Nicod has in mind Keynesian this score (more on that later). In any event, my main focus here will be on the [21] inductive probability (more on that later). While Nicod is not very clear on theory Hempel ultimately ends-up with. For present purposes, there is no need the logical details of his probability-raising account of instantial confirmation (stay to examine Hempel’s [17] ultimate theory of confirmation in its full logical glory. tuned for both Hempelian and Bayesian precisifications/reconstructions of Nicod’s For our subsequent discussion of Goodman’s “New Riddle,” we’ll mainly need just remarks), three aspects of Nicod’s conception of confirmation are apparent: the following two properties of Hempel’s confirmation relation, which is a relation Instantial confirmation is a relation between singular and general proposi- between statements and statements, as opposed to objects and statements.3 • tions/statements (or, if you will, between “facts” and “laws”).

1 Date: 01/02/08. Penultimate draft (final version to appear in the Journal of Philosophical Logic). Traditionally, such conditions are called “Nicod’s Criteria”, which explains the abbreviations. I would like to thank audiences at the Mathematical Methods in Philosophy Workshop in Banff, the Historically, however, it would be more accurate to call them “Keynes’s Criteria” (KC), since Keynes Reasoning about Probabilities & Probabilistic Reasoning conference in Amsterdam, the Why Formal [21] was really the first to explicitly endorse the three conditions (above) that Nicod later championed. 2 Epistemology? conference in Norman, the philosophy departments of New York University and the Not everyone in the literature accepts (EQCH ), especially if (EQCH ) is understood in terms of University of Southern California, and the Group in Logic and the Methodology of Science in Berkeley classical logical equivalence. See [28] for discussion. I will simply assume (EQCH ) throughout my for useful discussions concerning the central arguments of this paper. On an individual level, I am discussion. I will not defend this assumption (or its evidential counterpart, which will be discussed indebted to Johan van Benthem, Darren Bradley, Fabrizio Cariani, David Chalmers, Kenny Easwaran, below), which I take to be common ground in the present historical dialectic. 3 Hartry Field, Alan Hájek, Jim Hawthorne, Chris Hitchcock, Paul Horwich, Franz Huber, Jim Joyce, (NC) is clearly a more charitable reconstruction of Nicod’s (3) than (NC0) is. So, perhaps it’s most Matt Kotzen, Jon Kvanvig, Steve Leeds, Hannes Leitgeb, John MacFarlane, Jim Pryor, Susanna Rinard, charitable to read Hempel as taking (NC) as his reconstruction of Nicod’s remarks on instantial confir- Jake Ross, Sherri Roush, Gerhard Schurz, Elliott Sober, Michael Strevens, Barry Stroud, Scott Stur- mation. In any case, Hempel’s reconstruction still ignores the probabilistic aspect of Nicod’s account. geon, Brian Talbot, Mike Titelbaum, Tim Williamson, Gideon Yaffe, and an anonymous referee for the This will be important later, when we discuss probabilistic approaches to Goodman’s “New Riddle”. Journal of Philosophical Logic for comments and suggestions about various versions of this paper. At that time, we will see an even more charitable (Bayesian) reconstruction of Nicod’s remarks. GOODMAN’S “NEW RIDDLE” 3 4 BRANDEN FITELSON

(NC) For all constants x and for all (independent) predicate expressions φ, ψ: (H2) All emeralds are grue. More formally, H2 is: ( x)(Ex Gx). ∀ ⊃ [φx & ψx\ confirms [( y)(φy ψy)\ and Even more precisely, H2 is: ( x)[Ex (Ox Gx)]. ∀ ⊃ ∀ ⊃ ≡ [φx & ψx\ disconfirms [( y)(φy ψy)\. ∼ ∀ ⊃ And, we can state three salient singular (evidential) statements, about an object a: (M) For all constants x, for all (consistent) predicate expressions , , and for φ ψ ( 1) Ea & Ga.[a is an emerald and a is green] all statements H: If [φx\ confirms H, then [φx & ψx\ confirms H. E ( 2) Ea & (Oa Ga).[a is an emerald and a is grue] As we will see, it is far from obvious whether (NC) and (M) are (intuitively) correct E ≡ ( ) Ea & Oa & Ga.[a is an emerald and a is both green and grue] confirmation-theoretic principles (especially, from a probabilistic point of view!). E Severally, (NC) and (M) have some rather undesirable consequences, and jointly Now, we are in a position to look more carefully at the various claims Goodman they face the full brunt of Goodman’s “New Riddle,” to which I now turn. makes in the above passage. At the beginning of the passage, Goodman considers an object a which is examined before t, an emerald, and green. He correctly points 2. History: Hempel & Goodman out that all three of 1, 2, and are true of such an object a. And, he is also E E E Generally, Goodman [14] speaks rather highly of Hempel’s theory of confirma- correct when he points out that — according to Hempel’s theory of confirmation — 1 confirms H1 and 2 confirms H2. This just follows from (NC). Then, he seems tion. However, Goodman [14, ch. 3] thinks his “New Riddle of Induction” shows E E that Hempel’s theory is in need of rather serious revision/augmentation. Here is to suggest that it therefore follows that “the observation of a at t” confirms both an extended quote from Goodman, which describes his “New Riddle” in detail: H1 and H2 “equally.” It is important to note that this claim of Goodman’s can only be correct if “the observation of at ” is a statement (since Hempel’s rela- Now let me introduce another predicate less familiar than “green”. It is a t tion is a relation between statements and statements, not between “observations” the predicate “grue” and it applies to all things examined before t just in case they are green but to other things just in case they are blue. Then at and statements) which bears the Hempelian confirmation relation to both universal time t we have, for each evidence statement asserting that a given emerald statements H1 and H2. Which instantial, evidential statement can play this role? is green, a parallel evidence statement asserting that that emerald is grue. Neither 1 nor 2 can, since Hempel’s theory entails neither that 1 confirms H2 E E E And the statements that emerald a is grue, that emerald b is grue, and nor that 2 confirms H1. However, the stronger claim can play this role, since: so on, will each confirm the general hypothesis that all emeralds are grue. E E ( ) confirms H1 and confirms H2. Thus according to our definition, the prediction that all emeralds subse- † E E quently examined will be green and the prediction that all will be grue are That Hempel’s theory entails ( ) is a consequence of (NC), (M), and the following † alike confirmed by evidence statements describing the same observations. evidential equivalence condition, which is also a consequence of Hempel’s theory: But if an emerald subsequently examined is grue, it is blue and hence not (EQCE) If E confirms H, then anything logically equivalent to E also confirms H. green. Thus although we are well aware which of the two incompatible Figure 1, below, contains a visually perspicuous proof of ( ) from (NC), (M), and predictions is genuinely confirmed, they are equally well confirmed ac- † cording to our present definition. Moreover, it is clear that if we simply (EQCE). The single arrows in the figure are confirmation relations (annotated on choose an appropriate predicate, then on the basis of these same observa- the right with justifications), and the double arrows are entailment relations. tions we shall have equal confirmation, by our definition, for any predic- tion whatever about other emeralds—or indeed about anything else. . . . We ( x)(Ex Gx) ( x)[Ex (Ox Gx)] are left . . . with the intolerable result that anything confirms anything. ∀ ⊃ ∀ ⊃ ≡ Before reconstructing the various (three, to be exact) arguments against Hempel’s (NC) theory of confirmation that are implicit in this passage (and the rest of the chap- Ea & Ga Ea & (Oa Ga) ter), I will first introduce some notation, to help us keep track of the logic of the ↑ ≡ arguments. Let Ox x is examined before t, Gx x is green, Ex x is an Ö Ö 4 Ö (M) emerald. Now, Goodman defines “grue” as follows: Gx Ox Gx. With these Ö ≡ definitions in hand, we can state the two salient universal statements (hypotheses): (Ea &⇑Ga) & Oa (Ea & (Oa Ga)) & Oa (H1) All emeralds are green. More formally, H1 is: ( x)(Ex Gx). ≡ ∀ ⊃ (EQCE) 4Here, “ ” is the material biconditional. I am thus glossing the detail that unexamined-before- ≡ t grue emeralds are blue on Goodman’s official definition. This is okay, since all that’s needed for Ea & Oa & Ga ! Goodman’s “New Riddle” (at least, for its most important aspects) is the assumption that unexamined- = E before-t grue emeralds are non-green. Quine [26] would have objected to this reconstruction on the Figure 1. A Proof of ( ) from (NC), (M), and (EQC ). grounds that “non-green” is not a natural kind term (and therefore not “projectible”). Goodman did † E not agree with Quine on this score. As such, there is no real loss of generality here, from Goodman’s point of view. In any event, I could relax this simplification, but that would just (unnecessarily) com- Of course, the following weaker claim follows immediately from ( ): plicate most of my subsequent arguments. In the final section (on Goodman’s “triviality argument”), I † will return to this “blue vs non-green” issue (since in that context, the difference becomes important). ( ) confirms H1 if and only if confirms H2. ‡ E E GOODMAN’S “NEW RIDDLE” 5 6 BRANDEN FITELSON

Goodman is not only attributing the qualitative confirmation claim ( ) to Hempel’s a bridge principle connecting the logical relation of confirmation with the epistemic ‡ theory. He is also attributing the following quantitative (or comparative) claim: relation of evidential support.8 Here, it is helpful to consider an analogy.

(?) confirms H1 and H2 equally. A (naïve) “relevance logician” might be tempted to offer the classical logician the E 9 Let’s call Goodman’s argument concerning ( ) his qualitative “grue” argument, and following “reductio” of classical deductive logic (viz., a “paradox of entailment”): ‡ the argument concerning (?) his quantitative argument. There is also a third argu- (1) For all sets of statements X and all statements p, if X is logically inconsis- ment toward the end of the above passage, which is Goodman’s triviality argument. tent, then p is a logical consequence of X. I will spend most of the rest of the paper discussing the qualitative argument, 5 (2) If an agent S’s belief set B entails a proposition p (and S knows that B p), which I take to be the most important and interesting argument of the three. Be- 10 î then it would be reasonable for S to infer/believe p. cause I am most interested how Goodman’s argument bears on probabilistic con- (3) Even if S knows that their belief set B is inconsistent (and, on this basis, ceptions of confirmation, the remainder of the paper will largely be concerned with they also know that B p, for any p), there are (nonetheless) some p’s such Bayesian confirmation theory (of various flavors). After giving careful Hempelian î that it is not the case that it would be reasonable for S to infer/believe p. and Bayesian reconstructions of Goodman’s qualitative argument in section 3, I will return to the quantiative argument (this time, just from a Bayesian point of (4) Therefore, since (1)–(3) lead to absurdity, our initial assumption (1) must view) in section 4. In section 5, I will conclude with a (more brief) discussion of have been false — reductio of the classical “explosion” rule (1). Goodman’s triviality argument, from both Hempelian and Bayesian perspectives. Of course, a classical logician need not reject (1) in response to this argument. As Harman’s [15] and Macfarlane’s [22] discussions suggest, the classical logician may 3. Goodman’s Qualitative Argument instead choose to reject premise (2). It is premise (2) that plays the role of bridge 3.1. Prelude: Logic vs Epistemology in Goodman’s Qualitative Argument. Good- principle, connecting the logical concept (entailment) and the epistemic concept (in- man is right, of course, that ( ) is entailed by Hempel’s theory of confirmation.6 ference, or reasonable belief) that feature in the relevantist’s argument. Without ‡ And, Goodman seems to think that ( ) is intuitively false. Specifically, he seems to such a bridge principle, there is no way to get classical logic to have the unde- ‡ think that confirms H1 but does not confirm H2. As such, even this first argu- sirable “epistemic consequences” that seem to be worrying the relevantist. The E E ment (which comes before the quantitative and triviality arguments) is supposed Harman-esque strategy here for responding to the relevantist is to argue that (2) is to show that Hempel’s theory of confirmation has some unintuitive consequences. implausible, in precisely the sort of cases that are at issue in the argument. If an On reflection, however, it’s not clear precisely what is supposed to be unintuitive agent’s beliefs B are inconsistent (and they know this), the proper response may about ( ). After all, Hempel’s confirmation relation is a logical relation. Is Good- be to reject some of the members of B, rather than inferring anything (further) ‡ man’s worry here really a logical one? It seems to me that Goodman’s worry here is from B. For the remainder of the paper, I will use this Harman-esque approach as really epistemic, rather than logical.7 That is, I take it that Goodman’s intuition here a guiding analogy. In fact, I will argue that this strategy is in some ways even more is really that is evidence for H1 for epistemic agents S in “grue” contexts, but effective in the inductive case, for two reasons. First, in the inductive case, the wor- E E is not evidence for H2 for epistemic agents S in “grue” contexts. Notice that this ries are even more clearly epistemic in nature. And, second, there is a sense (to be intuition is about an epistemic relation (evidential support) in “grue” contexts, not explained below) in which Goodman’s qualitative argument (against Bayesianism) a logical relation (the Hempelian confirmation relation). Understood in this way, relies even more heavily on a bridge principle that the Bayesians should reject for Goodman’s argument is enthymematic — it’s missing a crucial premise to serve as independent reasons. Before applying this analogy to the Bayesian case (which will be my main focus here), I will begin with a brief discussion of what form it might 5In his initial paper on “grue” [13], Goodman’s arguments do not involve universal hypotheses at all (i.e., in [13], Goodman’s discussion is purely instantial). I will not discuss those early purely instantial arguments here (although, in my discussion of his later “triviality” argument in the final section of this paper, some instantial confirmation-theoretic considerations will arise). I’m focusing 8I borrow the term “bridge principle” from John MacFarlane [22], who also takes an epistemic here on Goodman’s later arguments (which happen to primarily involve universal hypotheses) for stance with respect to arguments of this kind. In the case of deductive logic (the so-called “paradoxes two reasons. First, Goodman is much clearer in [14] about his methodology and aims, i.e., on what of entailment”), MacFarlane [22] offers epistemic reconstructions much like the ones I am using here. the argument is supposed to show and how it is supposed to show it (see fn. 7). Second, the purely 9Of course, relevantists don’t like reductio arguments, but the aim of this argument is to convince instantial arguments in [13] are flawed in some elementary ways that have nothing to do with where a classical logician who does accept reductio. I’ll assume this is dialectically kosher. I think the deep and important problems with Goodman’s underlying methodology and aims lie. 10Note: this is just a “straw man” bridge principle. The point of the deductive example is merely to 6 As for (?), Hempel’s theory [17] does not have quantitative or comparative consequences, so it set-up an analogy with the inductive case. Of course, to make the deductive analysis more interesting, is neutral on (?). However, it can be shown that Hempel & Oppenheim’s quantitative extension of we’d have to consider much more subtle bridge principles [22]. That’s not the point of this paper. Hempel’s qualitative theory [19] does imply (?). I omit the details of this in order to conserve space. Where we will need to take greater care is in the inductive case. That’s what we aim to do, below. That But, I will analyze (?) from a Bayesian point of view (in some detail) in section 4, below. said, I should also note that in contexts where S’s beliefs are inconsistent, I doubt that any deductive 7Goodman [14, pp. 70–72] makes it quite clear that what’s at stake here are the principles of bridge principle (no matter how sophisticated) will serve the relevantist’s purposes. Specifically, I inductive logic (in the sense intended by Hempel). But, Goodman’s methodology for rejecting such suspect the relevantist faces a dilemma: Any bridge principle will either be false (while perhaps being principles is one that appeals to epistemic intuitions. This is most clear at [14, pp. 72–73], when strong enough to make their reductio classically valid), or it will be too weak to make their reductio Goodman discusses an example that is a prelude to “grue”. There, he says (concerning an instantial classically valid (while perhaps being true). The naïve bridge principle (2) stated above falls under the E and a universal H) that E “does not increase the credibility of” H “and so does not confirm” H. first horn of this dilemma. More plausible bridge principles will (I bet) not yield a valid reductio. GOODMAN’S “NEW RIDDLE” 7 8 BRANDEN FITELSON take with respect to Goodman’s qualitative argument against Hempelian confir- relation can be understood as three-place as well, but it would not make a substan- mation theory. Here is an analogous (albeit somewhat sketchy) reconstruction of tive difference to his theory. This is because Hempel’s theory (which is based on Goodman’s qualitative argument against Hempelian confirmation theory: entailment properties of E) has no way to distinguish the following two types [6]:

(10) Hempel’s theory of confirmation [specifically, property ( )]. Conditional Confirmation: E confirms H, relative to K. ‡ • Conjunctive, Unconditional Confirmation: E & K confirms H (relative to ). (20) Some bridge principle connecting Hempelian confirmation and evidential • > support. Presumably, it will say something like: if E Hempel-confirms H, Bayesian confirmation theory, which — unlike its Hempelian predecessor — is non- 11 then (ceteris paribus ) E is evidence for H for S in . monotonic in both E and K (more on this below), treats these two types of confirma- C tion claims in radically different ways. This is why its confirmation relation must (30) is evidence for H1 for an agent S in a “grue” context, but is not evidence E E be defined as explicitly three-place. Before we give a precise, Bayesian reconstruc- for H2 for an agent S in a “grue” context. tion of Goodman’s qualitative argument, it’s worthwhile to reconsider the crucial (4 ) Therefore, since (1 )–(3 ) lead to absurdity, our initial assumption (1 ) must 0 0 0 0 (controversial) Hempelian conditions (NC) and (M), from a Bayesian perspective. have been false — reductio of Hempelian inductive logic (1 )/( ). 0 ‡ Perhaps a Hempelian could try to save their confirmation theory by blaming (20) 3.3. The Hempelian Properties (NC) and (M) from a Bayesian Point of View. In a for the unintuitive (apparent) “epistemic consequences” brought out by Good- Bayesian framework, neither (NC) nor (M) is generally true. Let’s focus on (NC) first. man’s qualitative argument. In the end, I don’t think this Harman-esque strat- I.J. Good [11] was the first to describe background corpora K relative to which (e.g.) 12 egy is very promising for the Hempelian confirmation theorist. But, I won’t Ea & Ga does not confirm ( x)(Ex Gx). Here’s a Good-style example: ∀ ⊃ dwell on the (dim) Hempelian prospects of responding to Goodman’s qualitative Let the background corpus K be: Exactly one of the following two argument, Harman-style. Instead, I will focus on Goodman’s qualitative argu- hypotheses is true: (H1) there are 100 green emeralds, no non- ment, as reconstructed from a Bayesian point of view. As we will see shortly, all green emeralds, and 1 million other things in the universe [viz., Bayesian confirmation-theorists face a powerful (prima facie) qualitative challenge ( x)(Ex Gx)], or ( H1) there are 1,000 green emeralds, 1 blue from Goodman, but one that can be parried (or so I will argue) Harman-style. ∀ ⊃ ∼ emerald, and 1 million other things in the universe. And, let 1 E Ö & , where is an object randomly sampled from universe. 3.2. The Bayesian Conception of Confirmation — Broadly Construed. Like Nicod, Ea Ga a contemporary Bayesians explicate confirmation in terms of probabilistic relevance. In such a case (assuming a standard random sampling model Pr): But, unlike Nicod, most contemporary Bayesians (a notable exception being Patrick 100 1000 Pr( 1 H1 & K) < Pr( 1 H1 & K) Maher [23]) don’t use inductive (or logical) probabilities in their explication of con- E | = 1000100 1001001 = E | ∼ firmation. Rather, contemporary Bayesians tend to use credences or epistemically So, in such a context, the “positive instance” Ea&Ga actually disconfirms rational degrees of belief, which are assumed to obey the probability calculus. As it the corresponding universal generalization ( x)(Ex Gx). turns out, this difference makes no difference for Goodman’s qualitative argument ∀ ⊃ This seems to show that (NC) is false, from a Bayesian point of view. But, in (properly reconstructed), since the qualitative argument will go through for any Hempel’s response to Good’s paper [18], he explains that (NC) was always meant to probability function whatsoever (independently of its origin or interpretation). As 13 be interpreted in a way that would rule-out such corpora as illegitimate. Specifi- such, for the present discussion (of the qualitative argument ), I will work with the K cally, Hempel [18] suggests that the proper understanding of (NC) is as follows: following very broad characterization of Bayesian confirmation theory, which will apply to basically any approach that uses probabilistic relevance of any kind (this (NC ) For all constants x and for all (independent) predicate expressions φ, ψ: > includes just about every plausible flavor that’s been on the market since Keynes). [φx & ψx\ confirms [( y)(φy ψy)\ relative to a priori corpus K , and ∀ ⊃ > [φx & ψx\ disconfirms [( y)(φy ψy)\, relative to K . E confirms H, relative to K iff Pr(H E & K) > Pr(H K), where Pr( ) is ∼ ∀ ⊃ > | | · | · some “confirmation-theory suitable” conditional probability function. The idea behind (NC ) is that the salient confirmation relations here are supposed > Notice how the Bayesian confirmation relation is explicitly three-place (at least, to be relations between evidence and hypothesis in the absence of any other (em- once a “suitable” Pr is fixed), whereas Hempel’s was two-place. In fact, Hempel’s pirical) evidence. And, since Good’s example presupposes a substantial amount of other (background) empirical evidence, this is supposed to undermine Good’s ex-

11Depending on one’s epistemological scruples, one will require various ceteris paribus conditions ample as a counterexample to (NC). I have discussed this dialectic between Hempel to be satisfied here. For instance, one might require that S knows that E Hempel-confirms H in , and Good elsewhere [6], and I won’t dwell on it too much here. To make a long story C and/or that S possesses E as evidence in , etc. I will say more about such ceteris paribus conditions short, Hempel’s (NC )-challenge to Good has recently been met by Patrick Maher, C > on the Bayesian side, below. This sketchy Hempelian story is just a prelude to the Bayesian analysis. who shows [23] that even (NC ) is not generally true from a Bayesian perspec- 12 > This is because (a) Hempel does not even seem to realize he needs a bridge principle in order tive. Maher’s counterexample to (NC ), which makes use of Carnapian inductive to apply his confirmation theory, and (as a result) there is little indication as to how such a prin- > probabilities, is similar in structure to an example described by Good [12] in his ciple would or should have been formulated, and (b) Hempel’s theory entails (M), which makes the formulation of any remotely plausible bridge principles quite difficult (by Hempel’s own lights [6]). original response to Hempel. The details of this dispute about (NC) and (NC ) are > 13As we’ll see below, even the quantitative argument is less sensitive to the choice of confirmation- not important for our present purposes. What’s important for us is that (NC) is theoretic probability function Pr than one might have thought (or than some seem to think). not sacrosanct from a Bayesian point of view. Specifically, the two crucial Bayesian GOODMAN’S “NEW RIDDLE” 9 10 BRANDEN FITELSON morals here are: (i) whether a “positive instance” confirms a universal generaliza- universal generalizations [( y)(φy ψy)\. From a Bayesian point of view, how- ∀ ⊃ tion will depend sensitively on the background corpus relative to which it is taken, ever, only the conditionals are guaranteed to be confirming instances of universal 14 and (ii) “positive instances” need not confirm universal generalizations even rela- generalizations, relative to all background corpora K and all (regular ) probability tive to “tautological” or “a priori” background corpora. I will return to these crucial functions Pr. This is because (i) universal claims entail their conditional-instances, morals shortly, when I discuss Goodman’s claim ( ). But, first, I will briefly discuss but they do not entail their “conjunction-instances”, and (ii) if H entails E, then H ‡ the (im)plausibility of (M), from a Bayesian point of view. and E will be positively correlated under any (regular) probability function. This The Hempelian monotonicity property (M) is even more suspect than (NC). In- explains why it is possible to generate Good-style counterexamples to (NC). If by deed, Hempel himself had some compelling intuitions about confirmation that ran “positive instance” the historical commentators on confirmation theory had meant counter to (M)! In his original resolution (or dissolution) of the raven paradox [17] conditional statements of the form [φa ψa\, then no Bayesian counterexamples ⊃ (and in his response to Good [18], discussed above), Hempel appeals to an intu- to (NC) would have been (logically) possible. This shows just how important (and itive feature of confirmation (or inductive support) that directly contradicts (M). I strong!) assumption (M) is. Assumption (M) is so strong that the conjunction of (M) have discussed this inconsistency in Hempel’s writings on the raven paradox in and (NC) is equivalent to the conjunction of (M) and the following, innocuous (from detail elsewhere [6]. The salient point for our present purposes is that there are a Bayesian stance) instantial principle of confirmational relevance: reasons to reject (M) that are independent of Goodman’s “New Riddle” — reasons (NC ) For all constants x and for all (independent) predicate expressions φ, ψ: ⊃ that Hempel himself would have seen as compelling. Moreover, as in the case of [φx ψx\ confirms [( y)(φy ψy)\, and ⊃ ∀ ⊃ 15 (NC), there are also compelling Bayesian reasons for rejecting (M). Bayesian coun- [φx & ψx\ disconfirms [( y)(φy ψy)\. ∼ ∀ ⊃ terexamples to (M) are quite easy to come by. And, to my mind, counterexamples I suspect that some of the perplexity raised by Goodman’s discussion arises out of to (M) are much simpler and more compelling than counterexamples to instantial unclarity about precisely what a “positive instance” is. And, I think this unclarity in principles, such as (NC). Here’s one: let Bx x is a black card, let Ax x is the ace Ö Ö the writings of Hempel and Goodman is caused mainly by their implicit acceptance of spades, and let Jx x is the jack of clubs. Assuming (K) that we sample a card Ö of evidential montonicity (M), which obscures the distinction between conditional- a at random from a standard deck (thus, we take Pr to be the standard probability instances vs conjunctive-instances of universal statements.16 Be that as it may, function for such a model), we have the following probabilistic facts: let’s continue on with our (Bayesian) scrutiny of Goodman’s claim ( ). 1 1 ‡ Ba confirms Aa, relative to K, since Pr(Aa Ba & K) 26 > 52 Pr(Aa K). • | = = | 3.4.2. Bayesian Scrutiny of Goodman’s Claim ( ). Goodman’s claim ( ) says that Ba & Ja disconfirms (indeed, refutes!) Aa, relative to K, since: ‡ ‡ E confirms H1 iff confirms H2. While it is true that Hempel’s theory entails ( ), it • E ‡ 1 is not true that Bayesian confirmation theory — broadly construed — entails ( ). Pr(Aa Ba & Ja & K) 0 < Pr(Aa K) ‡ | = 52 = | In fact, using a technique similar to I.J. Good’s [11], we can describe background It seems pretty clear (and nowadays uncontroversial) that (M) is an undesirable corpora K (and probability functions Pr) relative to which confirms H1, but E E property for a confirmation relation to have (and Hempel himself was sensitive to disconfirms H2. Here is one such Good-style, Bayesian counter-example to ( ): ‡ this in his discussions of the raven paradox [6]). Moreover, (M) is highly implausible Let the background corpus K be: Exactly one of the following two for reasons that seem to be independent of Goodman’s “New Riddle”. hypotheses is true: (H1) there are 1000 green emeralds 900 of In the next section, I will discuss Goodman’s central qualitative confirmation- which have been examined before t, no non-green emeralds, and theoretic claim ( ), from a Bayesian point of view. 1 million other things in the universe [viz., ( x)(Ex Gx)], or ‡ ∀ ⊃ ( ) there are 100 green emeralds that have been examined before 3.4. Goodman’s Claim ( ) from a Bayesian Point of View. H2 ‡ t, no green emeralds that have not been examined before t, 900 3.4.1. What are “Positive Instances” Anyway? Before we start thinking about Good- 14 man’s ( ) from a Bayesian perspective, it is useful to re-think the notion of “positive Here, I am assuming that all evidential and hypothetical statements (all E’s and H’s) receive ‡ instance” that has been at work (historically) in this instantial confirmation liter- non-extreme unconditional probability on all salient probability functions (a reasonable assumption here). Hereafter, I will assume that we’re talking about regular probability functions in this sense. ature. So far, we have been talking about conjunctions [ & \ as “positive φa ψa The importance of this regularity assumption will become clear when we get to “old evidence”, below. instances” of universal generalizations [( y)(φy ψy)\. But, from a logical 15 ∀ ⊃ Notice how Nicod’s three principles are consequences of Bayesian confirmation theory, if “pos- point of view, instances of [( y)(φy ψy)\ are conditionals [φa ψa\. So, itive instance” is understood in the conditional sense (as opposed to the conjunction sense). This ∀ ⊃ ⊃ let’s add the two salient conditional claims to our stock of singular statements: provides a much more charitable reading of Nicod (as a Bayesian) than Hempel’s reconstruction of Nicod (as a deductivist). Moreover, these considerations provide further reason to worry about (M). ( 3) Ea Ga. [either a is not an emerald or a is green] E ⊃ 16My conjecture is that this insidious adherence to (M) is a vestige of Hempel’s uncharitable ( 4) Ea (Oa Ga). [either a is not an emerald or a is grue] “objectual” reconstruction (NC0) of Nicod’s instantial confirmation principle. If one thinks of objects E ⊃ ≡ or observations (rather than statements) as doing the confirming, one can easily be led into thinking From a logical point of view, then, it is 3 not 1 that is an instance of H1, and it is E E that (M) makes sense. For instance, one might start saying things like “observations of green emeralds 4 not 2 that is an instance of H2. Interestingly, Hempel’s theory also entails that E E are observations of emeralds, therefore anything confirmed by the observation that a is an emerald 3 confirms H1 and that 4 confirms H2. So, from a Hempelian point of view, both E E is also confirmed by the observation that a is a green emerald.” Such (monotonic) inferences are conjunctions [φa & ψa\ and conditionals [φa ψa\ are confirming instances of (intuitively) not cogent in general. Fallacies like this underlie many errors in confirmation theory. ⊃ GOODMAN’S “NEW RIDDLE” 11 12 BRANDEN FITELSON

non-green emeralds that have not been examined before t, and 1 based on “inductive” or “logical” probabilities.20 In this sense, Goodman’s argu- million other things in the universe (viz., ( x)[Ex (Ox Gx)]). ment seems to be even weaker than the relevantist’s (even if we replace the “straw ∀ ⊃ ≡ 17 Let Ea&Oa&Ga, for a randomly sampled from the universe. man” bridge principle (2) with a real contender), since it will have to lean more EÖ heavily on its “bridge principle”. In any event, as we’ll see next, even if ( ) were a In such a case (assuming a suitable sampling model Pr — see fn. 17): ‡ consequence of such a theory, Goodman’s qualitative argument would remain un- sound from a “logic-Bayesian” (or any other Bayesian) point of view, because it (still) 900 100 rests on an independently implausible bridge principle. That brings us (finally) to Pr( H1 & K) > Pr( H2 & K) E | = 1001000 1001000 = E | our (Harman-style) Bayesian reconstruction of Goodman’s qualitative argument. 3.5. A Proper Bayesian Reconstruction of Goodman’s Qualitative Argument. Be- fore we give a precise reconstruction of Goodman’s qualitative argument against Since H1 and H2 are mutually exclusive and exhaustive (given K), it Bayesian confirmation theory (in “reductio” form, as above), we need one more in- follows that confirms H1, but disconfirms H2 (relative to K). E gredient — a bridge principle to connect the formal/logical concept of confirmation and the epistemic concept of evidential support. The requisite bridge principle can Of course, all this shows is that there is a possible background corpus K/probability be found in Carnap’s work on applied inductive logic. Carnap was an advocate of function Pr relative to which confirms H1 and disconfirms H2. That is enough to E show that (unlike Hempel’s theory) Bayesian confirmation theory — broadly con- what we might call logic-Bayesianism. Like Hempel, Carnap thought that confir- strued — does not entail ( ). But, it is also true that Bayesian confirmation theory mation (now, understood as probabilistic relevance) was a logical relation between ‡ statements. Unlike Hempel, Carnap realized that in order to apply confirmation in is consistent with ( ). As we will see shortly, there are also corpora K/probability ‡ actual epistemic contexts , we need a bridge principle that tells us when confir- functions Pr relative to which does confirm H1 if and only if it confirms H2. C E matory statements constitute evidence for hypotheses (for agents S in ). Carnap At this point in the dialectic, a Hempelian might implore the Bayesian (as Hempel C [18] did to Good [11], in the context of the raven paradox) to determine whether or [2] discusses several bridge principles of this kind. The most well-known of these (and the one needed for Goodman’s argument) is stated by Carnap as follows: not ( ) is true relative to a priori background corpus K and “a priori” probability ‡ > function Pr . But, only those Bayesians (like Patrick Maher [23]) who believe in The Requirement of Total Evidence. In the application of inductive logic > inductive probabilities (as Nicod did) are likely to take such a challenge seriously. to a given knowledge situation, the total evidence available must be taken Most modern Bayesians do not have a theory of “confirmation relative to a priori as a basis for determining the degree of confirmation. background” to appeal to here.18 But, historically — i.e., at the time Hempel and Most modern Bayesians have (usually only implicitly, and without argument) ac- Goodman were thinking about “grue” — almost everyone who used probability to cepted the following precisification of Carnap’s requirement of total evidence: explicate the concept of confirmation used inductive or logical probability (as op- (RTE) E evidentially supports H for an epistemic agent S in a context if and only 21 C posed to subjective or epistemic probability) for this explication. So, it is important if E confirms H, relative to K, where K is S’s total evidence in . C to try to work out what the Bayesian confirmation relations look like, when induc- This principle (RTE) plays a crucial role in Goodman’s qualitative argument against tive probabilities are used in “impoverished” Goodmanian contexts (i.e., assuming a 19 Bayesian confirmation theory, which we can now carefully reconstruct, as follows: priori background K ). Interestingly, like (NC ), ( ) is not a consequence of (later > > ‡ (i) E confirms H, relative to K iff Pr(H E&K) > Pr(H K), for some “suitable” Pr. Carnapian) confirmation theories based on salient accounts of “inductive” or “log- | | [The “reductio” assumption: the formal Bayesian account of confirmation.] ical” probability. By emulating the structure of my Good-style counterexample to ( ) above in later Carnapian models (with three families of predicates), it can be (ii) E evidentially supports H for an epistemic agent S in a context if and only ‡ C shown that ( ) is not a consequence even of a “Carnapian” theory of confirmation, if E confirms H, relative to K, where K is S’s total evidence in . [(RTE)] ‡ C (iii) Agents S in Goodmanian “grue” contexts have Oa as part of their total CG evidence in .[i.e., S’s total evidence in (K) entails Oa.] CG CG 20This is a subtle issue, since Maher [24] argues that all of the existing Carnapian theories of inductive probability for three families of predicates (viz., for multiple properties) are inadequate. As such, Maher probably wouldn’t endorse such Carnapian counterexamples to ( ). However, the 17 ‡ Here, to make things concrete, assume the following. For each object in the universe, there is a existence of these Carnapian counterexamples to ( ) seems to have little or nothing to do with the ‡ slip of paper on which is inscribed a true description of the object in terms of whether it exemplifies reasons why Maher rejects the latest Carnapian theories of inductive probability for multiple prop- the properties E, O, and G. Imagine an urn containing all the slips. Assume a standard random erties. Moreover, there are intuitive analogues of these formal Carnapian counterexamples to ( ) [for ‡ sampling model Pr of the process, and suppose the slip that reads “Ea & Oa & Ga” is thus sampled. instance, the Good-style counterexample to ( ) that I describe above]. I think these considerations 18 ‡ Although Timothy Williamson does propose something like this in his [29, ch. 9]. See below. have the effect of shifting the burden of proof back to the Hempelian/Goodmanian defender of ( ). 19 21 ‡ The early Carnapian systems of [2] and [3] do entail (NC ) and ( ). However, the later Carnapian I will not offer an analysis of “S’s total evidence in ” here. One might think of this as S’s > ‡ C systems (e.g., those discussed in [23] and [24]) do not entail (NC ) or ( ). Maher’s [23] counterexam- background knowledge in , where this is distinct from the evidence S possesses in . Or, one might > ‡ C C ple shows that (NC ) can fail in later Carnapian systems. And, examples similar to my Good-style follow Williamson [29] and identify an agent’s knowledge in with the evidence they possess in . > C C counterexample to ( ) above can be emulated in later (e.g., [24]) Carnapian systems (details omitted). I’ll remain neutral on this question here, but I’ll stick with Carnap’s locution (for historical reasons). ‡ GOODMAN’S “NEW RIDDLE” 13 14 BRANDEN FITELSON

22 (iv) If K Oa, then—ceteris paribus — confirms H1 relative to K iff con- ways of responding to the old evidence problem have been floated in the literature î E E firms H2 relative to K, for any Pr-function. [i.e., if K Oa, then ( ).] [5]. And, one might hope that some of these responses to old evidence could be ap- î ‡ plied to “grue”. Unfortunately, most of the existing responses to the old evidence (v) Therefore, evidentially supports H1 for S in if and only if eviden- E CG E problem will not help with “grue”. Something more radical is needed. What we tially supports H2 for S in . CG need is a reformulation of (RTE) that avoids “grue”, old evidence, and other related (vi) evidentially supports H1 for S in , but does not evidentially support E CG E problems in one fell swoop. One approach that is suggested by the work of Carnap H2 for S in . CG [2, p. 472] and Williamson [29, ch. 9] is to radically reformulate the (RTE), and to Therefore, since (i)–(vi) lead to absurdity, our assumption (i) must be false revert to a priori background/probability function K /Pr in the definition of con- • > > — reductio of Bayesian confirmation theory (broadly construed)? firmation itself. This would bring confirmation theory back to something more like This argument is valid. So, let’s examine each of its premises, working backward the original (two-place) ideas that Nicod and Hempel had. Carnap called this sort from (vi) to (ii). Premise (vi) is based on Goodman’s epistemic intuition that, in of confirmation relation initial confirmation, which I will state as follows.25 “Grue” contexts, evidentially supports H1 but not H2. I will just grant this as- E E initially-confirms H iff Pr (H E & K ) > Pr (H K ), where Pr ( ) is sumption here (although, I think it could be questioned). Premise (v) follows log- > | > > | > > · | · some “a priori” conditional probability function, and K is a priori. ically from premises (i)–(iv). Premise (iv) is a theorem of the probability calculus. > Premise (iii) is an assumption about the agent’s background knowledge that’s im- Williamson suggests the following radical reformulation of (RTE) to go along with 26 plicit in Goodman’s set-up. Premise (ii) is (RTE). It’s the bridge principle, akin to (2) (his flavor of ) the Carnapian initial-confirmation relation: in the relevantists’ “reductio”. This is the premise on which I will focus. (RTE ) E evidentially supports H for S in iff S possesses E (as evidence) in , > C C Intuitively, some precisification of Carnap’s requirement of total evidence should and E initially-confirms H. be correct. The question at hand, however, is whether (RTE) is correct.23 While Richard Fumerton [9] discusses a similar approach, but he (being an epistemo- many Bayesians have implicitly assumed (RTE), there are good reasons to think it logical internalist, unlike Williamson) also requires that S have some “epistemic must be rejected by Bayesian confirmation-theorists — reasons that are indepen- access” to E, and to the fact that E initially-confirms H. These are fascinating re- dent of “grue” considerations. There are various reasons for Bayesian confirmation- cent developments in formal epistemology, but (owing to considerations of space) theorists to reject (RTE). I will only discuss what I take to be the most important I won’t be able to dwell on them further here. of these here. As Tim Williamson [29, ch. 9] explicitly points out, (RTE) must be Another approach (similar to some subjective Bayesian responses to the old ev- rejected by Bayesian confirmation theorists because of the problem of old evidence idence problem) would be to maintain the original Bayesian explication of confir- [10, 5]. This (now infamous) problem arises whenever is already part of (entailed E mation (as inherently three-place, where the “suitable” probability function is just by) S’s total evidence in . In such a case, we will have Pr(E K) 1, for any Pr. C | = the agent’s credence function at the time they are assessing the evidential support As a result, will be powerless to confirm any , relative to — for any flavor E H K relations in question), and non-radically reformulate (RTE) so that it doesn’t rela- of Bayesian confirmation theory. Thus, all Bayesian confirmation theorists must tivize to all of the agent’s background knowledge K in . The tricky thing about reject (RTE), on pain of having to say something absurd: that nothing that is part C this sort of response is that it is very difficult to specify a proper part K0 K of S’s of ’s total evidence can evidentially support any for . From this perspective, it ï S H S total evidence in that can avoid all the varieties of confirmational indistinguisha- can now be seen that Goodman’s (qualitative) “grue” problem and the old evidence C bility that one finds in the literature. One can think of all of these approaches as problem are related in the following way. In both “grue” and old-evidence contexts groping toward something like the following (seemingly useless) “bridge principle”: , the agent’s total evidence K is such that some undesirable evidential support C 24 (RTE ) E evidentially supports H for S in iff E confirms H, relative to K , where relation is entailed (assuming only logic and probability theory) by (RTE). Various 0 C 0 K is the logically strongest statement such that: (a) S’s total evidence in 0 C 22 entails K [K K ], and (b) no undesirable evidential support relations are Strictly speaking, premise (iv) also requires the following additional ceteris paribus condition: 0 î 0 Pr(Ea H1 & K) Pr(Ea H2 & K). This assumption is standardly made by Bayesians who analyze derivable from (RTE0) using logic and probability theory alone. | = | “grue” [27]. I will assume it also. I am focusing on the role that K Oa plays in the argument, since î I think that’s where the action is. Some Bayesians assume something stronger: K Ea & Oa [27]. I î resist making this additional assumption here, since (a) it forces ( ) to be true as well, and (b) it also of course) as cases of confirmation without support. In fact, however, this apparent asymmetry is † makes Goodman’s intuition (vi) seem far less intuitive (indeed, I would say it makes (vi) seem false). merely apparent, because the grue case can equally legitimately be described (assuming Goodman is 23 It is important to note that Carnap only endorsed (RTE) in connection with what he called right) as a case in which H2 is supported (by , for S, in ) but not confirmed (by , relative to K). 25 ∼ E CG E “confirmation as firmness” — a different conception of confirmation, according to which E confirms I am writing both Pr and K here to emphasize that nothing a posteriori (other than E) is > > H, relative to K iff Pr(H E & K) > r , for some threshold value r . Carnap never explicitly endorsed allowed to be conditionalized on, when we’re using Pr for confirmation-theoretic purposes. Here, I | > (RTE) for “confirmation as increase in firmness” (his name for the probabilistic relevance conception). am following Carnap, who used the symbol “t” rather than “K ” for similar purposes. > Indeed, Carnap never explicitly endorsed any precise bridge principle for the relevance conception. 26It may be somewhat unfair to saddle Williamson with a Carnapian view of the “suitable” proba- However, he did say some things that naturally suggest a different principle. I discuss this below. bility function. Perhaps Williamson’s “evidential probability function” is not exactly the sort of thing 24On the surface, it might seem as if old evidence cases and grue cases constitute counterexam- Carnap had in mind (e.g., it may be more dynamic and have a slightly more psychologistic flavor). In ples to different directions of (RTE). After all, old evidence cases are normally described as cases of any event, I think it’s more appropriate to lump Williamson in with this sort of approach, as opposed support without confirmation, but grue cases are normally described (assuming Goodman is right, to the sort of approach to old-evidence advocated by subjective Bayesians (discussed below). GOODMAN’S “NEW RIDDLE” 15 16 BRANDEN FITELSON

The project of formulating a useful rendition of (RTE0) is one of the “holy grails” For this reason, I will refer to the r -perspective on ( ) as the Likelihoodist perspec- of contemporary subjective Bayesian confirmation theory. I’m not optimistic about tive. From a Likelihoodist perspective, the questionC about Goodman’s (?) reduces the prospects for such an (RTE ). I think something more radical is needed. How- to a question about the likelihoods Pr( H1 & K) and Pr( H2 & K). And, once 0 E | E | ever, I also think that the Carnapian “initial confirmation” approach is unlikely to we assume (RTEc), ( ) is guaranteed — from a Likelihoodist perspective — by the succeed, because it is unlikely that any adequate theory of “logical” or “inductive” assumption that S’s total evidence in entails Oa. This was premise (iii) in the C C probability is forthcoming. I suspect that we need a different conception of in- qualitative argument, which I have already granted to Goodman as part of the set- ductive logic altogether. Once inductive logic is properly reformulated, a sensible up of his example. It is easy to show that (RTEc), (LL), and (iii) jointly entail ( ). This 28 alternative to (RTE), (RTE ) and (RTE0) may be possible. I have some ideas about is because, if K entails Oa, then (ceteris paribus ) Pr( H1 & K) Pr( HC2 & K). > 27 E | = E | how to begin such a project, but they are beyond the scope of this paper. So, Likelihoodists who adopt a naïve bridge principle such as (RTEc) are stuck with In the next section, I will examine Goodman’s quantitative argument [for (?)], Goodman’s quantitative claim ( ). Indeed, this is just the position that Sober [27] from a Bayesian perspective. — a Likelihoodist who also endorsesC (RTEc) — finds himself in. At this point, I could rehearse the various independent reasons (e.g., the old evidence problem) 4. Goodman’s Quantitative Argument why Bayesians should reject naïve bridge principles such as (RTEc). But, since I have already covered that issue in the qualitative case, I won’t (in the interest of 4.1. Goodman’s “Equal Confirmation” Claim. Goodman claims that formal theo- space) bother going over it again here (although, there are some other issues that ries of confirmation are stuck not only with ( ), but also with the implication that ‡ arise here, which are somewhat more subtle than the qualitative case). Instead, I confirms H1 and H2 “equally.” In this section, I will simply grant (arguendo) E will simply assume (for the rest of this section) that (RTEc) is adopted, and I will that ( ) is true, and I will focus on the (independent) plausibility of the “equally” ‡ focus on the measure-sensitivity of the above derivation of ( ), given (RTEc). claim. I will interpret Goodman’s “equally” claim as a comparative or relational Interestingly, not all Bayesian confirmation theorists areC Likelihoodists. As a confirmation claim. Specifically, I will read this “equal” claim as follows: result, even if one accepts (RTEc) and (iii), one need not accept ( ). Moreover, as I (?) c(H1, K) c(H2, K), where c(H, E K) is some Bayesian relevance have argued elsewhere [7], Likelihoodism (LL) has some undesirable consequences. E | = E | | C measure of the degree to which E confirms H, relative to K (c will be defined For instance, (LL) is incompatible with the following condition, which seems to me in terms of some “suitable” confirmation-theoretic probability function Pr). to be almost constitutive of the “provides better evidence for” (or “favors”) relation:

Of course, what’s really at issue here (as in the qualitative case above) is whether (F) If E provides conclusive evidential support for H1 for S in , but E provides C non-conclusive evidential support for H2 for S in , then E favors H1 over ( ) evidentially supports H1 and H2 equally strongly, for S in . 29 C E CG H2 for S in . As inC the qualitative case, ( ) cannot be established by confirmation theory alone. C I discuss this and other reasons for worrying about (LL) at length in my [7]. To make We also need a quantitative bridge principle, connecting quantitative confirmation, C a long story short, it turns out that, among all the historical Bayesian accounts and quantitative evidential support. Presumably, the naïve principle would be: of the favoring relation, only one is compatible with (F), assuming (RTEc). This (RTEc) The degree to which E evidentially supports H for S in is c(H, E K), C | leading rival to Likelihoodism is based on the likelihood-ratio measure l(H, E K) where c is some (appropriate) relevance measure of degree of confirmation Pr(E H&K) | = | , rather than the measure . Of course, the likelihood-ratio approach is Pr(E H&K) r (defined in terms of a “suitable” confirmation-theoretic probability function non-Likelihoodist|∼ , because it does not satisfy the “Law of Likelihood” (LL). But, the Pr), and K is S’s total evidence in . C l-based approach to confirmation satisfies the following, closely related condition: As above, it is only if we assume (RTEc) that ( ) can be established. Moreover, (LLF) If Pr(E H1 & K) Pr(E H2 & K) and Pr(E H1 & K) < Pr(E H2 & K), then whether ( ) can be so established will also depend on which measure c is used. | ≥ | 30 | ∼ | ∼ C c(H1,E K) > c(H2,E K). As it turns out, there are various Bayesian relevance measures c of degree of | | C What the -based approach to confirmation reveals, therefore, is that [assuming confirmation, and they tend to disagree radically on comparative (or relational) l (RTEc)] favoring relations depend not only on the likelihoods of the hypotheses claims of the form c(H1, K) c(H2, K). I have discussed this disagreement E | ≥ E | being contrasted Pr( H1 &K) and Pr( H2 &K), but also on the likelihoods of the in detail elsewhere [7]. Rather than attempting a general survey here, I will just E | E | negations of the hypotheses being contrasted Pr( H1 & K) and Pr( H2 & K). say something about how the two most popular Bayesian relevance measures (for E | ∼ E | ∼ comparisons of this kind) behave with respect to Goodman’s “New Riddle.” The 28 Pr(H E&K) As I explained above (fn. 22), this (strictly speaking) also requires the ceteris paribus condition: first measure I will discuss is the ratio measure r (H, E K) | . When Pr(H K) Pr(Ea H1 &K) Pr(Ea H2 &K), which is assumed by Bayesians who analyze “grue”. Many Bayesians | = | | = | it comes to relational confirmation claims of this kind, the behavior of r is very assume something stronger: K Ea & Oa, which entails Pr( H1 & K) Pr( H2 & K) 1 [27]. 29 î E | = E | = simple, because r satisfies the so-called “Law of Likelihood” (see [7] for discussion): In the background, here, I am assuming that if E entails H (and S knows this in , and S C possesses E in , etc.), then E provides conclusive evidence for H for S in . See [7] for discussion. In (LL) c(H1,E K) c(H2,E K) iff Pr(E H1 & K) Pr(E H2 & K). C C | ≥ | | ≥ | light of my present “Harmanian attitude,” I need to be very careful here about this sort of principle (which looks like a bridge principle Harman would reject!). For the purposes of the present discussion, 27In my [8], I sketch some preliminary ideas about how one might proceed here. I plan a much let’s assume that we’re not talking about contexts in which “paradox of entailment” type issues arise. 30 more extensive treatment of my preferred reformulation of inductive logic in a future monograph. Note: this is only a sufficient condition for favoring [assuming (RTEc)], not a necessary condition. GOODMAN’S “NEW RIDDLE” 17 18 BRANDEN FITELSON

I will call this second set of likelihoods catch-alls. Because of the dependence of theory of confirmation, ( ) leads to the absurd result that “anything confirms any- † favoring relations on catch-alls, a proper Bayesian analysis of Goodman’s claim ( ) thing”. The most charitable rendition of Goodman’s “triviality argument” that I requires not only checking to see if Pr( H1 & K) Pr( H2 & K), but also looking have been able to come up with has the following (high-level) form: E | = E | C at the relationship between the catch-alls Pr( H1 & K) and Pr( H2 & K) in E | ∼ E | ∼ x confirms both H1 and H2. [i.e.,( )] Goodmanian contexts. This means we must think about our negations: E † y confirms both Gb and Bb. [where b = “the first emerald examined ( H1) Some emeralds are non-green. ∴ E ∼ after t” (assuming one exists), and G and B are incompatible properties.] ( H2) Some emeralds are non-grue. That is, some emeralds are either (i) exam- ∼ z confirms anything. ined prior to t and non-green or (ii) unexamined prior to t and green. ∴ E { Mutatis mutandis, anything confirms anything. What if we could somehow motivate the claim that Pr( H1 &K) < Pr( H2 &K), E|∼ E|∼ ∴ for a “suitable” confirmation-theoretic probability function Pr? Then, so long as First, I will analyze this argument from a Hempelian point of view, and then from a we’re not Likelihoodists, we could argue that — even if Goodman’s qualitative Bayesian point of view. From a Hempelian point of view, premise (x) must be true, claim ( ) is granted — favors the green hypothesis over the “grue” hypothesis. since it is a logical consequence of Hempel’s theory. Having established that † E E While this wouldn’t undermine the main thrust of Goodman’s argument (which is confirms both that all emeralds are green and that all emeralds are grue, Goodman qualitative), it might be seen to “soften the impact” of the paradox by showing that says that this implies that confirms both that “the first emerald examined after t provides better evidence for H1 than it does for H2 in “grue” contexts. At this E E will be green” and that “the first emerald examined after t will be blue”. Let’s just point in the dialectic, those sympathetic to Goodman’s quantitative argument [27] assume (arguendo) that there will be a first emerald observed after t (and that it will be quick to point out that the only way we can have both 33 was not observed before t). Let’s call this emerald b. So, Goodman’s claim seems Pr( H1 & K) Pr( H2 & K), and to be that it is a consequence of Hempel’s theory that (y) confirms both that Gb • E | = E | E Pr( H1 & K) < Pr( H2 & K) and that Bb, where G and B are incompatible properties (say, Bx is equivalent to • E | ∼ E | ∼ & , for some ). This step from (x) to (y) cannot be correct, however, since is if Pr(H1 K) > Pr(H2 K) is also true. Indeed, this is just a theorem of the Gx φx φ | | ∼ probability calculus, and so it holds for all Pr. What this means is that any such (as Hempel proves in [16]) Hempel’s theory satisfies the consistency condition: non-Likelihoodist attempt to deny Goodman’s quantitative claim (in the “intuitive” (CC) If E confirms H1 and E confirms H2, then H1 and H2 are logically consistent direction) will boil-down to a claim that H1 is more probable “a priori” (prior to (provided only that E is self-consistent). ) than H2. As Sober [27] explains, this does not seem like a very satisfying way E of denying Goodman’s quantitative claim ( ). On such a story, the evidence is So, if isn’t true that Hempel’s theory implies that confirms both Gb and Bb E E not playing any real role in the differential confirmation that provides for H1 (where these are incompatible statements about an individual), then what is true? C E vs H2. What we seem to have here is simply some sort of “a priori” prejudice One “nearby” truth is that Hempel’s theory entails confirms both of the condi- 31 E against “grue” hypotheses enshrined in Bayesian formalism. This wouldn’t be tional claims Ob Gb and Ob Bb. Indeed, Hempel’s theory even entails that ∼ ⊃ 34 ∼ ⊃ much different (in spirit) than Goodman’s own “entrenchment” line on the paradox, confirms ( x)Ox! And, while it might be tempting to infer from this that the E ∀ and it wouldn’t be very illuminating either. conjunction & Ob must therefore confirm both Gb and Bb, even this is not a E ∼ Having said all that, I should remind you that I have already argued that Bayesians should reject the (RTE) — in both its qualitative and quantitative forms32 — for reasons that are independent of “grue”. As such, while the above discussion of 33 the quantitative argument has some interesting wrinkles, I don’t think it takes any- If we stick with a definite description “the first emerald examined after t” rather than a constant thing away from the diagnosis/resolution I’ve already offered Bayesians above. In “b”, then this just makes Goodman’s “triviality argument” worse. This is because “the first emerald the next section, I will briefly discuss Goodman’s “triviality” argument — the third examined after t is green” and “the first emerald examined after t is non-green” aren’t even mutually exclusive (much less logical opposites). And, some sort of logical conflict is assumed here. See below. and last argument implicit in that infamous passage from Fact, Fiction & Forecast. 34 It is surely an odd consequence of Hempel’s theory that confirms ( x)Ox. But, it is not the E ∀ triviality goodman wants. Note: it would not be odd to say that Oa confirms ( x)Ox, since Oa is ∀ entailed by ( x)Ox. The problem is that Hempel’s theory implies (M), which forces him to say this 5. Goodman’s “Triviality” Argument ∀ same thing about any observation statement (i.e., any statement that entails Oa). (M) strikes again! Goodman does not stop with the two confirmation-theoretic claims ( ) and (?). Furthermore, because Hempel’s theory also implies (SCC) (see below) we can derive another “odd con- † sequence” from this one — that confirms any universal claim about unexamined objects. Moreover, He goes on (in the passage quoted above) to claim that, in the context of Hempel’s E this will apply to any observation statement of the form [Oa & φa\, for any predicate expression φ. And, so, we have the ultimate “odd consequence” of Hempel’s theory of confirmation: (OC) any obser- 31 In this sense, Goodman’s quantitative problem is far more difficult for the Bayesian to solve vation statement confirms any universal generalization about unexamined objects. However, (OC) does than the quantitative versions of the ravens paradox. See [6] for a novel quantiative approach to the not imply (nor is it true) that we can get simultaneous Hempel-confirmation of inconsistent claims. So, ravens paradox that is much more promising than anything one can say (quantitatively) about “grue”. we still do not get the triviality Goodman wanted. Note, also, that we can derive (OC) from the three 32 If one rejects (RTE), then one must also reject (RTEc). After all, the qualitative (positive) confir- properties (NC), (M), and (SCC) of Hempel’s theory. We do not need to fiddle with “grue-ifications” of mation relation just corresponds to some inequality constraint on c. Thus, (RTEc) entails (RTE). the antecedents of universal claims, such as Goodman’s “emerose” construction — see fn. 36. GOODMAN’S “NEW RIDDLE” 19 20 BRANDEN FITELSON

35 consequence of Hempel’s theory. Hence, Goodman’s “triviality” argument against is the jack of hearts. Assuming (K) a card a is sampled at random from a standard Hempel’s theory of confirmation seems to fall short of its mark right here.36 deck (and assuming that Pr obeys the standard random card draw model): Interestingly, however, this seems to be the only gap in Goodman’s “triviality” 1 1 Ra confirms Aa, relative to K, since Pr(Aa Ra&K) 26 > 52 Pr(Aa K). argument against Hempel’s theory. That is, all of the other steps in the argument • | = = | 1 1 Ra confirms Ja, relative to K, since Pr(Ja Ra & K) > Pr(Ja K). would go through, if only this one did. Specifically, the step from (y) to (z) is OK • | = 26 52 = | (for a Hempelian), since Hempel’s theory entails both of the following principles: Is this a bad thing? That is, is it a bad thing for a theory of confirmation to have the consequence that some evidential propositions confirm each of a pair of contradic- (&) If E confirms H1 and E confirms H2, then E confirms H1 & H2. tory hypotheses? It seems clear to me that the answer is “No!”. After all, we are (SCC) If E confirms H1, and H1 entails H2, then E confirms H2. often faced with multiple contradictory competing scientific hypotheses, and in at It is easy to see that (y), (&), and (SCC) jointly entail (z). And, once we have (z) least some of these cases we think some of our evidence confirms (even provides in place, we can then get something very close to ({) in the following way. For some positive evidential support for) more than one of these competing hypothe- any observation statement E of the form [Oa & φa\, for any predicate expres- ses at the same time. So, intuitively, it seems to me that (CC) is false. I think it sion φ, it will follow from Hempel’s theory that E confirms both [( x)φx\ and would be a bad thing if our confirmation theory allowed there to be cases in which ∀ [( x)(φx Ox)\. Then, by (x)–(z) mutatis mutandis, E will confirm anything. E confirms both H and H. But, neither the Hempelian nor the Bayesian theory of ∀ ≡ ∼ Thus, we have the claim that any observation statement confirms any hypothesis, confirmation violates the following weaker version of the consistency condition: which is (basically) what Goodman meant by ({). (CC0) If E confirms both H1 and H2, then H1 and H2 are not logical opposites. To sum up: there are two senses in which Goodman’s triviality argument “al- Moving on through the argument now, even if premise (y) were true in Goodma- most” works, when aimed against Hempel’s theory of confirmation. First, except nian contexts (from a Bayesian point of view), (z) would not follow. But how can for Goodman’s (false) suggestion (y) that Hempel’s theory allows for simultaneous Bayesian confirmation theory manage to avoid the absurd consequence that some confirmation of inconsistent claims, the rest of his triviality argument concerning evidential statements will confirm every other statement? After all, if it’s possible Hempel’s theory goes through. Second, while his triviality argument fails to estab- for E to confirm both members of an inconsistent pair of statements, then won’t it lish the desired conclusion that every observation statement Hempel-confirms ev- follow that E will confirm anything, since anything follows logically from a logically ery statement, it can be shown that every observation statement Hempel-confirms inconsistent set? No. This reasoning presupposes the two principles (&) and (SCC) every universal statement about unexamined objects. However, this can be shown above. While (&) and (SCC) are true from a Hempelian point of view, they are false, in a more direct way than Goodman’s remarks suggest, simply by appealing to from a Bayesian point of view. Here’s a simple Bayesian counterexample to (SCC). properties (NC), (M), and (SCC) of Hempel’s theory (see fn. 34 and fn. 36). Let Rx x is a red card, let Ax x is the ace of diamonds, and let Sx x is some Ö Ö Ö From a Bayesian point of view, however, the “triviality” argument is a non- ace. Assuming (K) that we sample a card a at random from a standard deck: starter. First, as we have already discussed above, (x) is not a logical consequence 1 1 Ra confirms Aa, relative to K, since Pr(Aa Ra&K) > Pr(Aa K). of Bayesian confirmation theory (even in its “Carnapian” flavors). But, even if (x) • | = 26 52 = | 2 4 were true, premise (y) would not follow (for any Bayesian). Interestingly, however, Ra doesn’t confirm Sa, rel. to K, since Pr(Sa Ra&K) Pr(Sa K). • | = 26 = 52 = | this step is not blocked by (CC), since Bayesian confirmation theory does not sat- The ingredients of our counterexample to (CC) above can be used to construct a 37 isfy (CC). The following example shows that (CC) is false from a Bayesian point of counterexample to (&) in the following way. Let Rx x is a red card, let Ax x is Ö Ö view. Let Rx x is a red card, let Ax x is the ace of diamonds, and let Jx x the ace of diamonds, and let Jx x is the jack of hearts. Then, we have: Ö Ö Ö Ö 1 1 Ra confirms Aa, relative to K, since Pr(Aa Ra&K) > Pr(Aa K). • | = 26 52 = | 35 1 1 See Hooker’s excellent paper [20] for a very careful analysis, which explains why even this does Ra confirms Ja, relative to K, since Pr(Ja Ra & K) 26 > 52 Pr(Ja K). not follow in Hempel’s theory. I discovered Hooker’s paper after I had completed my own analysis • | = = | Ra does not confirm Aa & Ja, relative to K, since: of Goodman’s “triviality argument”. I highly recommend Hooker’s paper, which goes into great detail • about Goodman’s argument (and which seems not to be as widely read as it should be). I have omitted Pr(Aa & Ja Ra & K) 0 Pr(Aa & Ja K) lots of technincal details from my discussion here, since Hooker’s paper covers them all beautifully. | = = | 36Some of Goodman’s other remarks seem to suggest a different way of getting to (z), without Thus, while Goodman’s triviality argument does reveal several “odd conseqeunces” going through (y). In a footnote, Goodman introduces the predicate “emerose” (E). Let Rx x is a Ö of Hempel’s theory of confirmation, it is powerless, when aimed against Bayesian rose, and let our new predicate “emerose” be defined as follows: Ex (Ex & Ox) (Rx & Ox). Now, Ö ∨ ∼ confirmation theory. Indeed, Bayesian confirmation theory does not have any of let H be the hypothesis that all emeroses are grue [( x)(Ex Gx)]. It is a consequence of Hempel’s 0 ∀ ⊃ the odd consequences we saw (fns. 34 & 36) in connection with Hempel’s theory.38 theory that also confirms H . And, by fiddling with our antecedent and consequent predicates E 0 in clever ways such as these, we can get to Hempel-confirm all kinds of weird universal gener- E alizations. Specifically, we can get to Hempel-confirm any universal statement about unexamined 38For instance, it is not a consequence of Bayesian confirmation theory that (or any other ob- E E objects. But, since this does not imply Hempel-confirmation of incompatible claims about individual servation statement) confirms arbitrary universal generalizations about unexamined objects [it does unexamined objects, this just boils down to the “odd consequence” (OC) already discussed in fn. 34. not even follow from Bayesian confirmation theory that confirms ( x)Ox]. Moreover, using Good- E ∀ 37For a Bayesian, the step from (x) to (y) is blocked because Bayesian confirmation theory does man’s “emerose” construction (fn. 36) does not reveal any odd consequences of Bayesian confirmation not imply (SCC). See below. Since Hempel’s theory implies (SCC), he needs (CC) to block this step. theory either, since Bayesian theory (unlike Hempel’s) does not entail (NC), (M), or (SCC). GOODMAN’S “NEW RIDDLE” 21 22 BRANDEN FITELSON

6. Conclusion References

Goodman’s “New Riddle” does reveal some “odd” consequences of Hempel’s [1] A.J. Ayer, Probability & Evidence, Macmillan, London, 1972. confirmation theory (especially, in the “triviality” part of the argument — see [2] R. Carnap, Logical Foundations of Probability, Chicago University Press, Chicago, 1950 (second edition 1962). fn. 34). But, one does not need fancy “grue” predicates to cast serious doubt [3] , The Continuum of Inductive Methods, Chicago University Press, Chicago, 1952. 39 on the adequacy of Hempel’s theory. As we have seen, almost all of the “bad” [4] J. Earman, Bayes or bust? A critical examination of Bayesian confirmation theory, MIT Press, things Goodman (correctly) demonstrates concerning Hempel’s theory of confir- Cambridge, MA, 1992. mation can be established more directly, using only properties like (NC), (M), and [5] E. Eells, Bayesian problems of old evidence, in C. Wade Savage (ed.) Scientific theories, Minnesota (SCC). Indeed, I think (M) in particular is the ultimate cause of many of the problems Studies in the Philosophy of Science, Vol. X, University of Minnesota Press, Minneapolis, 205-223. [6] B. Fitelson, The paradox of confirmation, Philosophy Compass (B. Weatherson and C. Callender, Goodman and others have had with Hempel’s theory. Even Hempel himself would eds.), Blacwkell (online publication), Oxford, 2006, http://fitelson.org/ravens.htm. have recognized (M) as a shortcoming of his theory — if only it had been brought [7] , Likelihoodism, Bayesianism, and relational confirmation, Synthese, 2007. to his attention (see my [6] for more on this lacuna). Moreover, even though Good- http://fitelson.org/synthese.pdf. man’s arguments do reveal some “oddities” in Hempelian confirmation theory, they [8] , Logical foundations of evidential support, Philosophy of Science, 2007. do not reduce Hempel’s theory to triviality as Goodman suggested (see also [20]). http://fitelson.org/psa2004.pdf. [9] R. Fumerton, Metaepistemology and Skepticism, Rowman & Littlefield, 1995. On the other hand, if we adopt a proper Bayesian theory of confirmation — [10] C. Glymour, Theory and Evidence, Princeton University Press, Princeton, 1980. which is superior to Hempel’s theory in many ways that are independent of Good- [11] I.J. Good, The white shoe is a red herring, British J. for the Phil. of Sci. 17 (1967), 322. manian considerations — then Goodman’s “New Riddle” loses much of its force. [12] , The white shoe qua red herring is pink, British J. for the Phil. of Sci. 19 (1968), 156–157. Our analogy with the relevantists’ “reductio” of classical deductive logic suggests [13] N. Goodman, A Query on Confirmation, Journal of Philosophy, XLIII (1946), 383–385. that what Goodman’s qualitative and quantitative arguments really reveal is that [14] , Fact, Fiction, and Forecast, Harvard University Press, Cambridge, Mass., 1955. [15] G. Harman, Change in View, MIT Press, Cambridge: MA, 1986. Bayesians should not accept very naïve versions of Carnap’s requirement of total [16] C. Hempel, A purely syntactical definition of confirmation, J. of Symbolic Logic 8 (1943), 122–143. evidence like (RTE). This won’t come as a surprise to contemporary Bayesians, most [17] , Studies in the logic of confirmation, Mind 54 (1945), 1–26, 97–121. of whom have already seen independent reasons to reject (RTE), such as the prob- [18] , The white shoe: no red herring, British J. for the Phil. of Sci. 18 (1967), 239–240. lem of old evidence. Moreover, alternatives to (RTE) that can avoid both the old [19] C. Hempel and P. Oppenheim, A definition of “degree of confirmation”, Philosophy of Science, 12 evidence problem and the “grue” problem have been suggested both in Carnap’s (1945), 98–115. [20] C. Hooker, Goodman, ‘grue’ and Hempel, Philosophy of Science, 35 (1968), 232–247. work [2], and (more explicitly) in the recent formal epistemology literature (e.g., by [21] J. Keynes, A Treatise on Probability, Macmillan, London, 1921. Tim Williamson [29, ch. 9]). Finally, while Goodman’s “triviality” argument reveals [22] J. MacFarlane, In what sense (if any) is logic normative for thought?, manuscript, 2004. some rather odd properties of Hempel’s theory of confirmation (see fn. 34 and [23] P. Maher, Probability captures the logic of scientific confirmation, in Contemporary Debates in the fn. 36), it is simply a non-starter from a Bayesian (inductive-logical) point of view. Philosophy of Science (C. Hitchcock, ed.), Blackwell, Oxford, 2004. In this paper, I have left open (specifically) what I think Bayesian inductive logic [24] , Probabilities for multiple properties: the models of Hesse and Carnap and Kemeny, Erken- ntnis, 55 (2001), 183–216. (viz., confirmation theory) should look like, in light of Goodman’s “grue” prob- [25] J. Nicod, The logical problem of induction, (1923) in Geometry and Induction, University of Cali- lem. I have also left open (specifically) what I think Bayesians should say about fornia Press, Berkeley, California, 1970. the (epistemic) evidential support relations in “grue” contexts. I suspect that these [26] W.V.O. Quine, Natural kinds, Ontological Relativity and Other Essays, Columbia U. Press, 1969. omissions (especially, the epistemic ones) will be disappointing to some readers. [27] E. Sober, No model, no inference: A Bayesian primer on the grue problem, in Grue! The New My main aim in this paper has been merely to try to re-orient the dialectic concern- Riddle of Induction (D. Stalker ed.), Open Court, Chicago, 1994. [28] R. Sylvan and R. Nola, Confirmation without paradoxes, in Advances in Scientific Philosophy ing the bearing of Goodman’s “New Riddle” on the very possibility of (classical) (G. Schurz and G. Dorn, eds.), Rodopi, Amsterdam/Atlanta, 1991. inductive logic. Here, I have made use of an analogy with the “paradoxes of entail- [29] T. Williamson, Knowledge and its Limits, Oxford University Press, Oxford, 2000. ment”, and a “Harman-style” response to them. I have shown how an analogous Harman-style response to Goodman’s “grue” paradox (on behalf of Bayesian induc- University of California–Berkeley E-mail address: [email protected] tive logic/confirmation theory) might go. Not only does this analogy reveal a novel Bayesian response to Goodman’s arguments (as arguments against Bayesian induc- tive logic), it also suggests avenues for further elaboration and clarification of both the inductive-logical and the epistemic sides of Bayesianism. I plan to address both the inductive-logical and the epistemic issues raised by Goodman’s “New Riddle” more fully in a future monograph on pure and applied confirmation theory.

39“Grue” predicates are also not needed to establish cases of evidential underdetermination or confirmational indistinguishability of the kind supposedly embodied in Goodman’s ( ) and (?). Of † course, I’m not the first to make this point. A.J. Ayer [1, p. 84], and others, made this point long ago.